{% extends 'homepage.html' %}
{% block content %}

<p>The database consists of fields from three sources:
<ol>
<li>The PARI database from the Bordeaux PARI group
<li>Additional totally real fields of degrees from 6 to 10 computed by
  John Voight.
<li>Additional fields from John Jones-David Roberts database.
</ol>
</p>

<h3>Details of the fields contained in the database</h3>
<p>
<ol>
<li>PARI database: the database is complete for the discriminant ranges
shown.  It is possible that they are complete for larger
  ranges which depend on the signature, but this is not
  shown here.
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>degree</th>
<th>minimum discriminant</th>
<th>maximum discriminant</th>
</tr>
<tr class="{{ row_class.next() }}"><td>1<td>\(1\)<td>\(1\)</tr>
<tr class="{{ row_class.next() }}"><td>2<td>\(-10^6\)<td>\(10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>3<td>\(-10^6\)<td>\(2\cdot10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>4<td>\(-10^6\)<td>\(10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>5<td>\(-10^6\)<td>\(2\cdot10^7\)</tr>
<tr class="{{ row_class.next() }}"><td>6<td>\(-10^6\)<td>\(10^7\)</tr>
<tr class="{{ row_class.next() }}"><td>7<td>\(-11841551\)<td>\(149324209\)</tr>
</table>
</p>
<li>The Voight database is included and is complete for totally real
  fields in the following ranges.
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>degree</th>
<th>minimum discriminant</th>
<th>maximum discriminant</th>
</tr>
<tr class="{{ row_class.next() }}"><td>6<td>\(1\)<td>\(16771805\)</tr>
<tr class="{{ row_class.next() }}"><td>7<td>\(1\)<td>\(213873729\)</tr>
<tr class="{{ row_class.next() }}"><td>8<td>\(1\)<td>\(2556640000\)</tr>
<tr class="{{ row_class.next() }}"><td>9<td>\(1\)<td>\(25405254289\)</tr>
<tr class="{{ row_class.next() }}"><td>10<td>\(1\)<td>\(289254654976\)</tr>
</table>
</p>
<li>The Jones-Roberts database provides complete lists of fields satisfying
  a variety of conditions.
<p>
The degree of a field is given by \(n\).
<table>
{% set row_class = cycler("odd", "even") %}
<tr class="{{ row_class.next() }}"><td>Fields unramified outside \(\{2,3\}\)
 with \(n\leq 7\)
</tr>
<tr class="{{ row_class.next() }}"><td>Fields ramified at only one prime \(p\) with \(p<200\) with \(n\leq 9\) </tr>
<tr class="{{ row_class.next() }}"><td>Fields ramified at only two primes \(p\lt q \leq 5\) with \(n\leq 8\) </tr>
</table>
</p>
<p>
The remaining cases, the bound depends on the Galois group.  Galois groups
are given in the form \(n\)T\(t\) where \(n\) is the degree and \(t\)
it the T-number.  The bound \(B\) is for the root discriminant, so a
bound of \(B\) means the discriminant \(D\) satisfies \(|D|\leq B^n\).
</p>
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>\(n\)T\(t\)</th>
<th>\(B\)</th>
</tr>
<tr class="{{ row_class.next() }}"><td>7T3<td>\(26\)</tr>
<tr class="{{ row_class.next() }}"><td>7T5<td>\(38\)</tr>
<tr class="{{ row_class.next() }}"><td>8T3<td>\(20\)</tr>
<tr class="{{ row_class.next() }}"><td>8T5<td>\(50\)</tr>
<tr class="{{ row_class.next() }}"><td>8T15<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T18<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T22<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T26<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T29<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T32<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T34<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T36<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T39<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T41<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T45<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T46<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T2<td>\(20\)</tr>
<tr class="{{ row_class.next() }}"><td>9T5<td>\(20\)</tr>
<tr class="{{ row_class.next() }}"><td>9T6<td>\(20\)</tr>
<tr class="{{ row_class.next() }}"><td>9T7<td>\(30\)</tr>
<tr class="{{ row_class.next() }}"><td>9T7<td>\(30\)</tr>
<tr class="{{ row_class.next() }}"><td>9T8<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T12<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T13<td>\(12\)</tr>
<tr class="{{ row_class.next() }}"><td>9T14<td>\(18\)</tr>
<tr class="{{ row_class.next() }}"><td>9T15<td>\(18\)</tr>
<tr class="{{ row_class.next() }}"><td>9T16<td>\(12\)</tr>
<tr class="{{ row_class.next() }}"><td>9T17<td>\(18\)</tr>
<tr class="{{ row_class.next() }}"><td>9T18<td>\(12\)</tr>
<tr class="{{ row_class.next() }}"><td>9T19<td>\(18\)</tr>
<tr class="{{ row_class.next() }}"><td>9T21<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T23<td>\(17\)</tr>
<tr class="{{ row_class.next() }}"><td>9T24<td>\(12\)</tr>
<tr class="{{ row_class.next() }}"><td>9T25<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T26<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>9T29<td>\(10\)</tr>
<tr class="{{ row_class.next() }}"><td>9T30<td>\(10\)</tr>
<tr class="{{ row_class.next() }}"><td>9T31<td>\(10\)</tr>
</table>
</ol>
</p>

{% endblock %}
</html>
